Problem: Lisa owns a "Random Candy" vending machine, which is a machine that picks a candy out of an assortment in a random fashion. Lisa controls the probability of picking each candy. The machine is running out of "Honey Bunny," so Lisa wants to program it so that the probability of getting a candy other than "Honey Bunny" twice in a row is greater than $\dfrac{9}{4}$ times the probability of getting "Honey Bunny" in one try. Write an inequality that models the situation. Use $p$ to represent the probability of getting "Honey Bunny" in one try.
Solution: The strategy We know that Lisa wants to program her "Random Candy" machine so that the probability of getting a candy other than "Honey Bunny" twice in a row is greater than $\dfrac{9}{4}$ times the probability of getting "Honey Bunny" in one try. If we let $D$ denote the probability of getting a candy other than "Honey Bunny" twice in a row, we obtain the inequality $D>\dfrac94p$. Now, let's express $D$ in terms of $p$. Expressing the probability of getting candy other than "Honey Bunny" twice in a row We know that the probability of getting "Honey Bunny" in one try is $p$. Therefore, the probability of getting a candy other than "Honey Bunny" in one try is $(1-p)$. So the probability of getting a candy other than "Honey Bunny" twice in a row is $(1-p)\cdot (1-p)$ or $(1-p)^2$. Putting things together We found that $D=(1-p)^2$. Since $D>\dfrac94p$, we can substitute and find an inequality in terms of $p$ that models the situation. The answer is: $ (1-p)^2>\dfrac94p$